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Why can't Eisenstein's Criterion be used to show that $$4x^{10} - 9x^{3} + 21x - 18$$ is irreducible over $\mathbb{Q}$?

I mean even if we were to apply Eisenstein here, there doesn't exist a prime $p$ that would apply all the E.C. rules anyways.

A detailed explanation would be great! Thanks.

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  • $\begingroup$ @WillJagy but why would you want to do that when you have a degree as big as 10? $\endgroup$ – dean3794 May 13 '15 at 21:14
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    $\begingroup$ Your cuestión is a bit weird: Eisenstein's criterion cannot be applied to that polynomial simply because there is no prime for which the required condition is satisfied. $\endgroup$ – Mariano Suárez-Álvarez May 13 '15 at 21:39
  • $\begingroup$ @MarianoSuárez-Alvarez hmm okay. Because working this out as a true and false question o thought it meant that using the Eisenstein Method is not sufficient to show that this polynomial is irreducible $\endgroup$ – dean3794 May 13 '15 at 21:41
  • $\begingroup$ I'm not sure what your question is. Is it "here is a polynomial that I know to be irreducible, why can't I deduce that it's irreducible from Eisenstein?" because you seemed to have answered that already in the statement of your question - i.e. Eisenstein doesn't apply for any prime $p$, since $3^2 | 18$. $\endgroup$ – James May 13 '15 at 21:44
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If $f(x)$ is a polynomial in $\mathbf Z[x]$ such that $f(ax+b)$ is Eisenstein with respect to a prime $p$ for some integers (or rational numbers) $a$ and $b$, then (i) $f(x)$ is irreducible over the $p$-adic numbers $\mathbf Q_p$ and (ii) $p$ divides the discriminant of $f(x)$. I find with PARI that your polynomial has discriminant with prime factors $2$, $3$, $61$, $293$, and $50997533$, and for each of these primes $p$ I find with PARI that $f(x)$ is reducible over $\mathbf Q_p$ (it has a factor of degree $1$ or $2$ in each case). This proves you can't convert $f(x)$ into an Eisenstein polynomial by a linear change of variables.

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  • $\begingroup$ Hello. Off topic, but I wanted to know if you had a reference for Fermat's own proof of the two square theorem, as you stated here, as a remark on page 28? I couldn't find the entire proof anywhere. $\endgroup$ – Henry Dec 3 '18 at 16:44
  • $\begingroup$ @Henry there is no record of a detailed proof by Fermat on anything. He usually just claimed to be able to do things, and a few times he very briefly sketched the idea. For the two square theorem, see Remark 5.2 of math.uconn.edu/~kconrad/blurbs/ugradnumthy/descent.pdf. $\endgroup$ – KCd Dec 3 '18 at 19:58
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The only prime that divides 9, 21, and 18 is 3. But $3^2 | 18$, so the Eisenstein criterion does not apply here.

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    $\begingroup$ The polynomial $1+x+x^2+\dots+x^{p-1}$ is irreducible by Eisenestein's theorem. But your method doesn't works. $\endgroup$ – k1.M May 13 '15 at 21:23
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    $\begingroup$ Eisenstein applies there, but not directly. You must substitute $x\mapsto x+1$. In general, it is not easy to see whether or not this is possible, so I answered the question for the specific polynomial specified by the OP. $\endgroup$ – William Stagner May 13 '15 at 21:28
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    $\begingroup$ So you can't say that a method doesn't applies, because always there exists tricky methods, and this is mathematics... $\endgroup$ – k1.M May 13 '15 at 21:30

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